Ab-Initio Calculations of the Dissociation Energy and ... · Abstract. Molecular orbital...

JKAU: Sci., Vol. 21 No. 2, pp: 227-240 (2009 A.D. / 1430 A.H.); DOI: 10.4197 / Sci. 21-2.4

227

Ab-Initio Calculations of the Dissociation Energy and

Periodic Properties of the Heavy P-block Dimers

Salem A. Hameed

Chemistry Department, Faculty of Science,

King Abdulaziz University, Jeddah, Saudi Arabia

Abstract. Molecular orbital calculations within the ab-initio frame

work using SBK-basis set at the RHF level are reported for heavy P-

block dimers of the fourth (Ga2, Ge2, As2, Se2 and Br2), fifth (In2, Sn2,

Sb2, Te2 and I2) and sixth (Tl2, Pb2 and Bi2) rows. The results of the

molecular orbital interpreted and correlated in terms of equilibrium

bond length, bond order, bonded valence, total valence, total energy,

nuclear energy, electronic energy, electron-electron energy, electron-

nuclear energy, nuclear-nuclear energy and dissociation energy. The

effect of d-orbital on the ground state properties is also reported. The

results indicate that method used gives fairly satisfactory predication

of the molecular properties.

Introduction

The electronic and spectroscopic properties of small clusters of heavy

atoms and metal atoms have been the topics of many experimental[1-15]

and theoretical studies[16-31]

. Experimental progress in this area has been

phenomenal in recent years[32-43]

due to the advent of laser vaporization

and supersonic jet expansion method. Typically, a sample of a foil or

sheet of material containing these elements such as a Ga As crystal, a Si

crystal or Pt foil is vaporized with a high energy Nd:YAG laser and

passed through a supersonic nozzel. A number of other techniques such

as sputtering methods, rare-gas matrix isolation methods, particle

bombardment method, ligand-stripping methods and aggregation within

zeolites and molecular sieves have been used for cluster generation[44 -53]

.

S.A. Hameed

228

Clusters of heavy main group elements comprise a subject

themselves, a separate class of unusual compounds whose properties

exhibit phenomenal variation as a function of size. Many experimental

and theoretical investigation in recent years have demonstrated that the

geometries and electronic properties of small clusters containing

typically 3-10 atoms have no resemblance at all to the more familiar

properties of the bulk. It is this aspect that has made the cluster area

perhaps one of the most exciting and challenging areas for both

experimental and theoretical activities. Electron affinities and ionization

potentials of these species exhibit dramatic alternations as a function of

cluster size.

A quick review of the chemical literature, shows that no systematic

investigations of the electronic structure and bonding of heavy p-block

dimers have been undertaken. Such calculations, together with the

experimental of spectroscopic properties could provide a wealth of

information on the chemistry of such important class of compounds.

In the present study we reported the calculation of the ground state

electronic properties (bond lengths, bond orders, total energies, electronic

energies, nuclear energies and dissociation energy) of heavy p-block

dimers of the fourth, fifth and sixth rows of the periodic table and

correlation of periodic trend.

Method of Calculation

The tremendous cost of ab–initio calculations has motivated many

attempts to find computational shortcuts. One such approach is based on

the observation that core orbitals are relatively inert to changes in

chemical bonding (the so called frozen core approximation)[54]

. Another

observation is that the effect of core electrons on valence electrons can be

treated through the use of potential energy term expressed as sum of local

functions multiplied by the projection operators. Based on these two

assumptions effective core potentials (ECP,s) or pseudo potentials, as

they are sometimes called, reduce the computational problem to dealing

with valence electrons only. Early results obtained from ECP,s compared

favorably well with these obtained from all electron calculations. These

calculations trended, however, to yield shorter bond lengths and

somewhat deeper potential energy curves. Such problems have been

Ab-Initio Calculations of the Dissociation Energy and…

229

overcome, and calculations using ECP,s or all electrons results are in

good agreement.

All calculations were carried out using SBK (Stevens / Basch /

Krauss) valence basis set[55]

, as implemented in Gaussian 94[56]

. This

choice implies an unscaled 3-21G basis.

Results and Discussion

The data obtained using SBK-MO calculations are analysed through

three rows, fourth (Ga, Ge, As, Se and Br), fifth (In, Sn, Sb, Te and I)

and sixth (Tl, Pb and Bi) and five groups, group III (Ga, In and Tl), group

IV (Ge, Sn and Pb), group V (As, Sb and Bi), group VI (Se and Te) and

group VII (Br and I).

Table 1 presents molecular orbitals, molecular states, total energies

and dissociation energies of the studied dimers. Among the values given

in Table 1 one can reveal the followings:

1. Within fourth row, the most stable dimer formed is Ga2 as

indicated from the drop of the total energy. The order of stability of the

fourth row are Ga > Br2 > Se2 > As2 > Ge2. Hence the Ge2 dimer is the

lowest stable dimer in fourth row.

2. With the fifth row, again In2 is the most stable dimer whereas,

Sn2 dimer is the lowest stable one. The order of stability are In > I2 > Te2

> Sb2 > Sn2.

3. Due to the lack of experimental data in sixth row, three dimers

are considered Tl2, Pb2 and Bi2. The stability of Bi2 is greater than Pb2 by

about 3.9au.

4. Within the group III, Ga2 dimer still the most stable dimer of all

heavy p-block elements.

5. Within group IV, Ge2 is more stable than Sn2 and Pb2 whereas,

Sn2 and Pb2 have the same stability.

6. Within group V, As2 is the more stable than Sb2 and Bi2. The

order of stability are As2 > Bi2 > Sb2.

7. Within group VI, Se2 dimer is more stable than Te2 dimer by

about 2.55 au.

8. Within group VII, Br2 dimer is more stable than I2 dimer by

about 3.83 au.

9. Thermodynamic studies[57]

of Ga2 revealed that the dissociation

energy of Ga2 is about 1.4eV, whereas, the theoretical dissociation

S.A. Hameed

230

energy calculated at complete active space self consistent field

(CASSCF) is 1.2eV. The difference may be attributed to the using the

incorrect partition function of Ga2.

Table 1. Molecular orbitals, molecular states, total energies and dissociation energies of the

studied dimers using SBK-basis set.

D,eV ET , au

Calc. Exp. SBK+d SBK

Molec.

state MO – configuration Dimer

1.40 1.20 –514.21 –514.24 ∑+1

g gguσσσ

222

112 Ga2

2.65 2.29 –7.30 7.28 ∑−3

g

2

u

π2

1g

σ

2

2u

σ

2

1 g

σ 1 Ge2

3.96 2.71 –12.11 –12.06 ∑+1

g

4

u

π2

1

g

σ

2

2 u

σ

2

1 g

σ 1 As2

3.19 2.91 –18.27 –18.22 ∑3

g

2

g

π4

1 u

π2

1 g

σ

2

2 u

σ

2

1 g

σ 1

Se2

1.97 1.88 –26.28 –26.25 ∑+1

g

4

g

π4

1

u

π 2 1

g

σ 2 2

u

σ21

g

σ 1 Br2

0.87 0.83 –376.4 –376.45 4

3π

1

u

π1

1g

σ

2

2u

σ

2

1 g

σ 1 In2

1.94 1.86 –6.49 –6.66 ∑+3

g

2

u

π2

1g

σ

2

2u

σ

2

1 g

σ 1 Sn2

3.09 2.17 –10.59 –10.55 ∑+1

g

4

u

π2

1g

σ

2

2u

σ

2

1 g

σ 1 Sb2

2.68 1.69 –15.72 –15.69 ∑+3

g

2

g

π4

1u

π2

1 g

σ

2

2 u

σ

2

1 g

σ 1 Te2

2.68 2.06 –22.45 –22.42 ∑+1

g

4

g

π4

1

u

π

2

1g

σ

2

2u

σ

2

1 g

σ 1 I2

0.63 0.58 –343.41 –34.3.40 4

3π

1

u

π1

1g

σ

2

2u

σ

2

1 g

σ 1 Tl2

0.86 0.88 –6.67

–6.66 ∑3

g

2

u

π2

1g

σ

2

2u

σ

2

1 g

σ 1 Pb2

2.04 2.30 –10.63 –10.61 ∑+1

g 4

u

π2

1u

σ

2

2u

σ

2

1 g

σ 1

Bi2


231

10. The theoretical CASSCF De value for the ground state −

∑g

3 of

Ge2 is 2.29 eV. This is a resamable agreement with the experimental

value of 2.65 eV. The small discrepancy between the calculated and

observed values could be attributed to the high order correlation

correction as well as the effective core potentials.

11. The theoretical CASSCF dissociation energy of the ground state

of Se2 was found to be 2.91eV.A number of experimental De values were

obtained (3.41, 3.16 and 3.110eV)[58]

. Photoionization and thermo-

chemical studies[59]

seem to favor the higher value. Balasubramanian[60]

thus obtained an upper value for De, eliminating the highest value of the

three possible De values. Thus the De value should be about 3.19 eV.

12. The main effect of higher order correlations was found to be on

the dissociation energies of Sb2 dimer. The calculated De increased by

17% due to the higher order correlations. The refined theoretical De (2.17

eV) was still about 30% smaller than an experiment (thermodynamic)

value of 3.09eV obtained by Knudsen effusion mass spectrometric

method.

13. The theoretical dissociation energy De obtained with the

CASSCF method for Te2 dimer was found to be 1.69 eV. The spin-orbit

interaction decreased the De value, since the separated atoms are more

stabilized by the spin-orbit interaction compared to the molecule. The

experimental value reported for Te2 is 2.68eV based on a weighted mean

of a number of values obtained from spectroscopic and thermochemical

methods. The difference between theoretical and experimental may be

attributed to limitation of the calculations.

14. The theoretical De for Pb2 dimer is 0.88 eV was almost in exact

agreement with an experimental value of 0.86eV. The agreement of such

magnitude on a very heavy dimer such as Pb2, for which both spin-orbit

effects and relativistic effects are large, must be considered impressive.

Periodic Properties

Bond length, bond order, total valence, bonded valence, nuclear

energy, electronic energy, electron-electron repulsion energy, electron-

nuclear attraction energy and nuclear-nuclear repulsion energy are

presented and plotted in Tables 2-5 and Fig. 1-6. The periodicity of the

studied dimers explained within a row on going from left to right

(increasing of molecular weight) and in the group on going from up to

S.A. Hameed

232

down (increasing of outer orbital shells). Among the values given in

Tables 2-5 and Fig. 1-6 one can reveal the following:

Table 2. Bond length and bond order of the studied dimers.

Table 3. Total valence and bonded valence of the studied dimers.

Bond length ( Ao) Bond order Dimer

SBK SBK+d SBK SBK+d

Ga2

Ge2

As2

Se2

Br2

2.5771

2.2322

2.1163

2.2337

2.3920

2.5780

2.2004

2.0723

2.1527

2.2838

1.307

2.020

2.653

1.715

0.871

1.330

2.225

2.815

1.814

0.917

In2

Sn2

Sb2

Te2

I2

2.8881

2.5571

2.4722

2.6351

2.7700

2.9001

2.5382

2.4349

2.5654

2.6750

1.380

2.108

2.761

1.801

0.883

1.339

2.219

2.869

1.892

0.950

Tl2

Pb2

Bi2

3.0277

2.6424

2.5871

3.0344

2.6342

2.5645

1.173

1.925

2.763

1.158

2.013

2.849

Total valence Bonded valence

Dimer

SBK SBK+d SBK SBK+d

Ga2

Ge2

As2

Se2

Br2

1.307 1.330

2.020 2.225

2.653 2.815

1.715 1.814

0.871 0.917

1.307 1.330

2.020 2.225

2.653 2.815

1.715 1.814

0.871 0.917

In2

Sn2

Sb2

Te2

I2

1.380 1.339

2.108 2.219

2.761 2.869

1.801 1.892

0.883 0.950

1.380 1.339

2.108 2.219

2.761 2.869

1.801 1.892

0.883 0.950

Tl2

Pb2

Bi2

1.173 1.158

1.925 1.925

2.763 2.849

1.173 1.158

1.925 1.925

2.763 2.849


233

Table 4. Nuclear energy, electronic energy and total energy of the studied dimers.

Table 5. Electron-electron energy, electron- nuclear energy and nuclear-nuclear energy of

the studied dimers.

Nuclear energy Electronic energy Total energy Dimer

SBK SBK+d SBK SBK+d SBK SBK+d

Ga2

Ge2

As2

Se2

Br2

90.55 90.49

3.79 3.84

6.25 6.38

8.52 8.84

10.83 11.35

–604.79 –604.70

–11.07 –11.14

–18.31 –18.49

–26.75 –27.11

–37.08 –37.63

–514.24 –514.24

–7.28 –7.30

–12.06 –12.10

–18.22 –18.26

–26.24 –26.27

In2

Sn2

Sb2

Te2

I2

80.80 80.46

3.31 3.33

5.35 5.43

7.22 7.42

9.36 9.69

–457.25 –456.92

–9.79 –9.82

–15.90 –16.02

–22.91 –23.14

–31.78 –32.14

–376.45 –376.46

–6.47 –6.49

–10.55 –10.58

–15.68 –15.71

–22.42 –22.44

Tl2

Pb2

Bi2

77.07 76.90

3.20 3.21

5.11 5.15

–420.49 –420.32

–9.86 –9.88

–15.72 –15.78

–343.41 –343.41

–6.66 –6.67

–10.60 –10.62

E e – e E e – N E N – N

Dimer SBK SBK+d SBK SBK+d SBK SBK+d

Ga2

Ge2

As2

Se2

Br2

498.40 498.35

7.55 7.67

13.11 13.37

19.83 20.25

28.14 28.72

–1452.67 –1452.38

–21.20 –21.20

–35.62 –36.08

–52.83 –53.64

–74.00 –75.16

90.55 90.46

3.79 3.84

6.25 6.38

8.52 8.84

10.83 11.35

In2

Sn2

Sb2

Te2

I2

371.07 370.70

6.64 6.72

11.31 11.50

16.89 17.19

23.96 24.37

–970.27 –969.52

–18.47 –18.59

–30.33 –30.64

–44.44 –44.97

–62.14 –62.92

80.80 80.46

3.31 3.33

5.35 5.43

7.22 7.42

9.36 9.69

Tl2

Pb2

Bi2

337.48 337.32

6.55 6.60

11.05 11.18

–867.16 – 866.81

–18.50 –18.57

–29.90 –30.08

77.07 76.90

3.20 3.21

5.11 5.15

S.A. Hameed

234

Br

Se

As

Ge

Ga

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

130 140 150 160 170

Molecular weight

Bon

d leng

th

Ga

Ge

As

Se

Br

0

0.5

1

1.5

2

2.5

3

135 140 145 150 155 160 165

Molecular weight

bo

nd

ord

er

BrSe AsGe

Ga

-800

-600

-400

-200

0

130 140 150 160 170

Molecular weight

Ele

ctr

on

ic e

ne

rgy

Br SeAs

Ge

Ga

0

20

40

60

80

100

130 140 150 160 170

Molecular weight

Nu

cle

ar

en

erg

y

BrSeAsGe

Ga

-600

-500

-400

-300

-200

-100

0

130 140 150 160 170

Molecular weight

To

tal

en

erg

y

Se BrAsGe

Ga

-2000

-1500

-1000

-500

0

130 140 150 160 170

Molecular weight

E e

-N

SeBr

As

Ge

Ga

0

20

40

60

80

100

130 140 150 160 170

Molecular weight

E N

-N

Br SeAsGe

Ga

0

100

200

300

400

500

600

135 140 145 150 155 160 165

Molecular weight

E e

- e

Fig. 1. The variations of bond length, bond order, nuclear energy and electronic energy of

the fourth row with molecular weight.

Fig. 2. The variations of total energy, Ee – N, EN – N and Ee – e of the fourth row with

molecular weight.


235

I

Te

Sb Sn

In

2.4

2.5

2.6

2.7

2.8

2.9

3

225 235 245 255 265

Molecular weight

Bo

nd

le

ng

th

Te

I

Sb

Sn

In

0

0.5

1

1.5

2

2.5

3

220 230 240 250 260

Molecular weight

Bo

nd

ord

er

Te

ISb

Sn

In

0

20

40

60

80

100

220 230 240 250 260

Molecular weight

Nu

cle

ar

en

erg

y

TeISb Sn

In

-500

-400

-300

-200

-100

0

220 230 240 250 260

Molecular weight

Ele

ctr

on

ic e

nerg

y


the fifth row with molecular weight.

Fig. 4. The variations of total energy, Ee – N, EN – N and Ee – e of the fifth row with molecular

weight.

TeISb Sn

In

-400

-300

-200

-100

0

220 230 240 250 260

Molecular weight

To

tal en

erg

y

TeISb

Sn

In

0

20

40

60

80

100

220 230 240 250 260

Molecular weight

E N

-N

TeI

Sb Sn

In

0

100

200

300

400

220 230 240 250 260

Molecular weight

E e

-e

TeISbSn

In

-1200

-1000

-800

-600

-400

-200

0

220 230 240 250 260

Molecular weight

E e

-N

S.A. Hameed

236

Bi

Pb

Tl

2.5

2.6

2.7

2.8

2.9

3

3.1

405 410 415 420

Molecular weight

Bo

nd

le

ng

th

0

0.5

1

1.5

2

2.5

3

405 410 415 420

Molecular weight

Bo

nd

ord

er

-500

-400

-300

-200

-100

0

405 410 415 420

Molecular weight

Ele

ctr

on

ic e

ne

rgy

0

10

20

30

40

50

60

70

80

90

405 410 415 420

Molecular weight

Nu

cle

ar

en

erg

y

-400

-300

-200

-100

0

405 410 415 420

Molecular weight

To

tal en

erg

y

-1000

-800

-600

-400

-200

0

405 410 415 420

molecular weight

E e

-N

0

100

200

300

400

405 410 415 420

Molecular weight

E e

-e

0

20

40

60

80

100

405 410 415 420

Molecular weight

E N

-N


the sixth row with molecular weight.

Fig. 6. The variations of total energy, Ee – N, EN – N and Ee – e of the sixth row with molecular

weight.


237

1. The total energy of row IV dimers increases on going from left to

right, i.e. from Ga2 to Br2. The increase in energy are systematic in case

of Ge2, As2, Se2 and Br2 in which a single double and triple bonds are

formed. The order of stability follow Br2 > Se2 > As2 > Ge2.

2. In case of Ga2-dimer a dramatic change in the total energy

reaches up to ≈ 480 au i.e. the most stable dimer in row IV is Ga2.

3. The addition of d-orbitals as polarized function does not affect the

total energies of all dimers, this may be attributed to the filling of d-

orbitals.

4. The same trend were observed in case of electronic energy and

electron-nuclear attraction energy but it reverse in case of nuclear energy,

nuclear-nuclear repulsion and electron-electron repulsion energies. (c.f.

Fig. 1 and 2).

5. The bond length of fourth row heavy dimers varies in a

systematic way on going from left to right (i.e. increasing of molecular

weight). It is found that the smallest value is As2 (triple bond) and the

largest value is Ga2 (single bond).

6. The variation of bond lengths are supported by the variation of

the bond order values, As2 dimer have the largest bond order and Br2

dimer (single bond) have the smallest bond order. The same trend can be

observed in case of total valence and bonded valence.

7. The bond order, total and bonded valence are affected by the

addition of d-orbitals to the SBK-basis set (c.f. Tables 2-5).

8. The same trend observed in the fifth row dimers at which In2 is

the most stable dimer. The rest of the fifth row dimers follows the order

In2 > Te2 > Sb2 > Sn2.

9. The largest bond order dimer of fifth row is Sb2-dimer (smallest

bond length), this may be attributed to the formation of triple bond. On

the other hand, the smallest bond order is I2.

10. The variation of energy components of the fifth row follow the

same trend as that of the fourth row.

11. In case of the sixth row Tl2-dimer is the most stable one, whereas,

Bi2-dimer is more stable than Pb2-dimer by about 4au. Due to the lack of

experimental data and parametrization in SBK-basis set, the rest of sixth

row (Po and At) cannot be studied.

12. Tl2-dimer (single bond) is the largest bond length (smallest bond

order) whereas, Bi2-dimer (triple bond) is the smallest bond length

(largest bond order).

S.A. Hameed

238

13. Within the groups ( III → VII ) the stability as measured by the

total energy decreases on going from up to down. This may be attributed

to the increase of the total valence and bonded valence. Ga2-dimer is

more stable than In2 and Tl2-dimer by ≈ 138 and ≈ 171au respectively.

14. In case of group IV and V Ge2 and As2 are more stable than (Sn2

and Pb2) and (Sb2 and Bi2). Both (Sn2 and Pb2) and (Sb2 and Bi2) dimers

have the same stability.

15. In group VI and VII, Se2-dimer is more stable than Te2 and Br2

than I2 by about 3 au and 4 au respectively.

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S.A. Hameed

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